The degree of unsaturation, also called the index of hydrogen deficiency, counts how many rings plus pi bonds a molecule contains and is found from the formula $DoU=(2C+2-H)/2$ for a compound of carbon and hydrogen only. Substituting $C=6$ and $H=10$ gives $DoU=(2\times 6+2-10)/2=(12+2-10)/2=4/2=2$. So the molecule has two degrees of unsaturation, which could be two double bonds, one triple bond, one ring plus one double bond, or two rings. A value of $1$ is wrong because it would correspond to formula $C_6{H}_{12}$, which has two more hydrogens. A value of $3$ is incorrect since it would require formula $C_6{H}_8$ with two fewer hydrogens. A value of $0$ would describe a fully saturated acyclic alkane $C_6{H}_{14}$, which does not match the given formula. This is the standard NCERT and JEE method of relating formula to unsaturation. A plausibility check: each missing pair of hydrogens relative to the saturated alkane corresponds to one degree of unsaturation, and $C_6{H}_{14}$ minus $C_6{H}_{10}$ is four hydrogens, giving two.